​Editor: Haley Brandt
Notes Sections 4-7:Katie Conner
Example Problems: Matthew Aguilar
Definitions: Austin Gibbons
Note sections 1-3: Cheyenne Bohlen

1.1 Real Numbers and Number Operations:
Real numbers can be pictured as points on a line called a real number line.
<---------------------0------------------------>

the zero is the origin.

Properties:
closure- a + b= real number. ab= real number
commutative- a+b= b+a ab=ba
associative- (a+b) +c= a+(b+c) (ab)c= a(bc)
identity- a+0=a,0+a=a
inverse- a+(-a)=0
distributive- a(b+c)=ab +ac

1.2 Algebraic Expressions and Models
numerical expressions=consist of numbers, operations, and grouping symbols.
evaluating expressions-
1. do operations inside grouping symbols.
2. evaluate powers.
3. multiply and divide.
4. add and subtract.

1.3
an equation is a statement where two expressions are equal.
a linear equation can be written as ax=b.

Important Formulas to know!

To Find:	Use the Formula:	Variable Meanings:
Distance	d=rt	D=distance R=rate T=time
Simple Interest	I=Prt	R=rate I=interest P=principal T=time
Temperature in Fahrenheit	F=9/5C+32	F=Degrees Fahrenheit C=degrees Celsius
Area of a Triangle	A=1/2bh	A=area B=base H=height
Area of a Rectangle	A=lw	A=area L=length w=width
Perimeter of a Rectangle	P=2L+2W	P=perimeter L=length W=width
Area of a Trapezoid	A=1/2(b1+b2)h	A=area b1=base H=height b2=base 2
Area of a circle	A=(pi)rˆ2	A=area Pi=3.14 R=radius
Circumference of a circle	C=2(pi)r	C=circumference Pi=3.14 R=radius

Rewriting Equations:

How to solve for 'y': 3y-6x=18

1.Write the Equation: 3y-6x=18

2.Add 6x to each side:3y=6x+18

3.Divide by 3 on each side:y=2x+6

Calculating the Value of 'y' when 'x'=2:

1.Substitute 2 in for 'x': 2(2)+6=y

2.Multiply: 4+6=y

3.Add: 10

1.5 Problem Solving Using Algebraic Models:

Problem Solving Method:

Write a Verbal Model---->Assign Labels---->Write an Algebraic Model---->Solve Algebraic Model----> Answer the Question

Example:Solving for the Area of a Rectangle

1.Write Verbal Model: Area=Length*Width

2.Assign Labels: A=area L=6 Width=5

3.Write and Algebraic Model: A=6*5

4.Solve Algebraic Model: A=30

5.Answer the Question: The area of the rectangle is 30 units squared.

Real Life Application for Setting Up an Equation and Solving To Find a Desired Answer:

Assume you are in a foreign country (against your will) and you are left with only American currency.

You cannot exchange your American dollars for foreign tender, but you can use the money to directly purchase the necessities you might need to get back home.

One thing you have discovered is the approximate exchange rate between the American dollar and the foreign bill that is used in the country.

If eight American dollar equals 87 times the foreign tender, you would then have to calculate how many American dollars it would require to purchase a plane ticket that costs 696 of the foreign currency.

Explanation: To set up the equation, you must label each piece of information that is given. The exchange rate is 8:87. This represents the function, (f). The amount of American dollars that you would need to use is represented by the variable (x). The total cost in the foreign currency (696) is final represented by (y). Because you know all the variables' information except for (x), it is concluded that (x) is the variable to be solved for. The first step of setting the equation up is finding out which variables are related. Because the exchange rate between American to Foreign currency is 8 to 87, the two variables (f) and (x) must be decided to be related. That is the first half of the equation. The other half is the total of the desired plane ticket, (y). So the final equation would be (f)(x) = (y), or (87/8)x = 696. The reason (87/8) is used is because the exchange rate is being used in reverse, where you are not using the rate by multiplying it by the foreign currency to gain the total American amount, not initially at least. From here you can solve equation to discover how many dollars are needed to pay. The first step is to obviously multiply each side by (8/87) because it is the reciprocal of (87/8), which will allow you to get the variable (x) alone. The result of multiplying 696 by (8/87) is 64. So the final result of the equation is (x) = 64. This means it will require $64.00 to purchase a plane ticket out of the country.​

Definitions:

Verbal Model - a word equation that represents a real-life problem
Algebraic Expression - a mathematical statement that represents a real-life problem

1.6 Solving Linear inequalities:

Ways to keep inequalities equivalent:

1.Add the same number to both sides of the equation

2.Subtract the same number from both sides of the equation

3. Multiply the same positive number to both sides of the equation

4. Divide the same number from both sides of the equation

5. When multiplying or dividing by a negative number you must change the direction of the sign.

Graphing Inequalities:

Compound Inequality

1< x < 5

ß---○---|---|---|---○---à

……1…2…3…4…5

The numbers from the equation are marked with open circles because the sign doesn’t indicate that they are solutions to the problem. The numbers between the circles are bolded because they are all solutions to the equation.

“Or” Inequalities:
x >5 or x < 2

ß---|---•---|---|---•---|---à
……1…2…3…4…5…6
The numbers from the equation are marked with closed circles because the sign indicates that they are solutions to the problem. The numbers above 6 and below 2 are also bolded because they could be possible solutions to the problem.

Setting Up and Solving a Real Life Inequality:

Assume that you are responsible for estimating how much distance a jet will be able to cover from a single fill up.

You know that after several testing experiences that in cold, moist conditions with high turbulence, the distance is brought down to 264 miles per fill.
In addition to this knowledge, you also know that in dry, warm conditions with the assist of a tailwind, the mileage hits its peak at 747 miles.
The best way to represent this knowledge is through a compound inequality.
Only a single variable will be used due to the nature of the problem.

Explanation: First you must start off by assessing what information will go where. The minimum mileage is 264. This will be placed as the first third of the inequality. The maximum is 747 miles so this will be placed as the last third of the inequality. The final piece is represented by (x) because it is the mileage that is expected in neither of the other conditions. The way to set up the in equality is 264 < x < 747. "Less than or equal to" is used because the numbers used on each end are maximum and minimums that were replicated in tests, so they are possibilities in the mileage. This inequality can tell you how many fill-ups will be needed on a given destination if the flying conditions are also given. This is done by assessing the flying conditions to get the expected mileage, and then compare that mileage to the total destination distance to determine the number of fill-ups needed.

Definitions:

​Linear Inequality in one variable - an inequality such as 2n - 3 › 9, and the inequality symbol is placed between two expressions
Linear Inequality in two variables - an inequality that can be written in one of the following forms: Ax + By ‹ C, Ax + By (less than or equal to sign) C,
Ax + By › C, or Ax + By (greater than or equal to sign) C
Solution of an Inequality in one variable - a value of the variable that makes the inequality true
Solution of an Inequality in two variables - an ordered pair (x,y) that, when x and y are substituted in the inequality, gives a true statement
Compound Inequality - two simple inequalities joined by "and" or "or"
Graph of an Inequality in one variable - all points on a real number line that correspond to solutions of the inequality
Graph of an Inequality in two variables - the graph of all solutions of the inequality

1.7 Solving Absolute Value Equations and Inequalities:

Example of an "OR" Absolute Value Equation:

|ax+b|>c

How To Solve when A=2, B=-5, and C=7:

OR Equation:.. .|2x-5|>7 ----> 2x-5>7 OR 2x-5<-7

1.Add 5:.... 2x>12 OR.. 2x<-2 2.Divide by 2:... x>6 OR.... x<-1

Example of a Compound Absolute Value Equation:

|ax+b|<c

How To Solve when A=2 B=-5 and C=7:

Compound Equation: |2x-5|<7 ----> 2x-5<7 AND 2x-5>-7

1.Add 5: 2x<12 AND 2x>-2

2.Divide by 2: x<6 AND x>-1

Also written as: -1<x<6

Definitions:

​Absolute value of a real number - the distance the number is from 0 on a number line

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